Clausen's theorem and hypergeometric functions over finite fields

Ron Evans; John Greene

doi:10.1016/j.ffa.2008.09.001

Clausen's theorem and hypergeometric functions over finite fields

Ron Evans
, John Greene

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

26 Scopus citations

Abstract

We prove a general identity for a ₃F₂ hypergeometric function over a finite field F_q, where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen's theorem expressing a ₃F₂ as the square of a ₂F₁. As another application, we evaluate an infinite family of ₃F₂ (z) over F_q at z = - 1 / 8. This extends a result of Ono, who evaluated one of these ₃F₂ (- 1 / 8) in 1998, using elliptic curves.

Original language	English (US)
Pages (from-to)	97-109
Number of pages	13
Journal	Finite Fields and their Applications
Volume	15
Issue number	1
DOIs	https://doi.org/10.1016/j.ffa.2008.09.001
State	Published - Feb 2009

Keywords

Clausen's theorem
Gauss sums
Gegenbauer functions
Hypergeometric functions over finite fields
Jacobi sums

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10.1016/j.ffa.2008.09.001

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abstract = "We prove a general identity for a 3F2 hypergeometric function over a finite field Fq, where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen's theorem expressing a 3F2 as the square of a 2F1. As another application, we evaluate an infinite family of 3F2 (z) over Fq at z = - 1 / 8. This extends a result of Ono, who evaluated one of these 3F2 (- 1 / 8) in 1998, using elliptic curves.",

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N2 - We prove a general identity for a 3F2 hypergeometric function over a finite field Fq, where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen's theorem expressing a 3F2 as the square of a 2F1. As another application, we evaluate an infinite family of 3F2 (z) over Fq at z = - 1 / 8. This extends a result of Ono, who evaluated one of these 3F2 (- 1 / 8) in 1998, using elliptic curves.

AB - We prove a general identity for a 3F2 hypergeometric function over a finite field Fq, where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen's theorem expressing a 3F2 as the square of a 2F1. As another application, we evaluate an infinite family of 3F2 (z) over Fq at z = - 1 / 8. This extends a result of Ono, who evaluated one of these 3F2 (- 1 / 8) in 1998, using elliptic curves.

KW - Clausen's theorem

KW - Gauss sums

KW - Gegenbauer functions

KW - Hypergeometric functions over finite fields

KW - Jacobi sums

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Clausen's theorem and hypergeometric functions over finite fields

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